Quantifier Elimination for Formulas Constrained by Quadratic Equations via Slope Resultants

Abstract

An algorithm is given for eliminating the quantifier from a formula: (∃x∈R)[α 2 x 2 +α 1 z+α 0 =0ΛF], where F is a quantifier free formula in x 1 ,..., x r , x, and α 2 , α 1 , α 0 are polynomials in x 1 ,..., x r with real coefficients such that the system {α 2 =0, α 1 =0, α 0 =0} has no solution in R r . The output formulas are made of resultants and their variants, which we call slope resultants. The slope resultants can be, like the resultants, expressed as determinants of certain matrices